We investigate regularity functions corresponding to a fixed Lyapunov bundle E in the Oseledets multiplicative ergodic theorem for an ergodic flow or diffeomorphism. These functions measure the deviation of the growth in norm of the linearized system from constant exponential growth. In general, Oseledets regularity functions were only known to be Borel measurable. Our main result is that if E is continuous on the entire manifold, then the corresponding regularity functions are locally bounded on an open set of full measure. If the system is uniformly mixing, then they are in L p, for all 1 ≤ p < ∞.